Mathematical Analysis, An Introduction
Real Number system
Inductive set and the set of natural numbers N. Infinite set, countable set. Well ordering property of N. The integers and the rational numbers. The real numbers. The Field axioms, the positivity axioms, the archimedean property and the (order) completeness axiom. Supremum and infimum
Definition. Definition of convergence. Properties: sums, product and reciprocal. Comparison test. Squeeze Theorem. Convergence implies boundedness. Monotone convergence theorem. Cauchy sequence. Cauchy principle of convergence. Subsequences. Bolzano Weierstrass Theorem. Every real bounded sequence has a monotone subsequence. Sequence tending to + or - infinity. Subsets of the real numbers, the intervals. limit point or cluster point or accumulation point of a subset of R. Open and closed subsets. Closure. Sequentially compact subset of R. Heine Borel Theorem. Countable compactness and sequentially compactness.
ε - δ definition of continuity. Sequence definition of continuity. Properties: Sum, product and quotient. Composition. Consequence of continuity. Continuous image of compact subset is compact. Extreme Value Theorem. Maximizer and minimizer. Continuous image of an interval is an interval. Intermediate Value Theorem. Monotone functions and continuity Inverse function and continuity. Uniform continuity. Any continuous function on a closed and bounded interval is uniformly continuous. Limits of a function. Properties: sum, product and quotient. Composition and limit. One sided limits. Squeeze Theorem for limits of functions.
Definition of derivative. Properties: sum, product and quotient. Chain rule. Relative maximum and relative minimum. Local minimizer and local maximizer. Relative Extremum. Rolle's Theorem. Mean Value Theorem. Consequences of Mean Value Theorem. Monotone functions. First derivative test and second derivative test for relative extremum. Concavity. Concavity and derived function. Derivative of inverse function. Cauchy mean value Theorem. L' Hôpital's Rule and an analytic consequence. Taylor Theorem with remainder. Intermediate Value property of the derived function, Darboux theorem.
Anti-derivative. Properties: additivity. Change of variable for anti-derivative. Riemann sums and Riemann integral. Lower and Upper Darboux sums. Lower and upper integrals. Refinement Lemma. Convergence of Lower and upper Darboux sums. Equivalent definitions of Riemann integrability. Any continuous function on a closed and bounded interval is integrable. Additivity. Properties of integrable functions. Convergence of Riemann sums. Mean Value Theorem for integrals. Darboux fundamental theorem of calculus. First fundamental theorem of Calculus, Second fundamental theorem of calculus. Products and modulus of integral functions. Integration by parts. Change of variable formula for Remain integrals. Second Mean Value Theorem for integrals. Improper integrals, convergence, absolute convergence, conditional convergence, tests for convergence, Lebesgue integral and improper integral, differentiation under the integral sign, the probability integral.
There are 14 chapters with exercises. These have been used in a course in mathematical analysis. Below we give a reasonable lesson plan for self studies and included typical choice of tutorial questions.
The chapters are in pdf format.
Here is the content page Contents.
Chapters and references
Ref: Chapter 2 Dedekind's cuts
Chapter 6 Series
Chapter 7 Series of Functions and Power Series
Chapter 8 Uniform Convergence and Differentiation
Chapter 10 Weierstrass Approximation Theorem
Chapter 11 The Elementary Functions
Chapter 12 Arithmetic of Power Series
Chapter 13 Special Tests for Convergence
Chapter 14 Improper and Lebesgue Integral
|Week 1||From natural numbers to the real
numbers. Motivation for a definition of real numbers, from a logical
point of view and from a desirable point of view in terms of its
Counting numbers to integers - the need for zero
From integers to the rational numbers, its algeraic structures as a totally ordered field. The notion of positive cone, ordering on the rational numbers.
The meaning of irrational numbers. Dedekind cut as a possible definition. (Detail knowledge not required.)
Real numbers as a complete totally ordered field. Definition of upper and lower bounds, supremum and infimum. The completeness property. The archimedean property. The rational number field is not complete. Density of the rationals and also of the irrationals.
|Week 2||Sequences. Definition. Convergence of
sequence. Sequences other than real sequences, complex sequence, sequence
in Rn, etc. Equivalent definition of convergence for real sequence in
terms of continuity. Convergence of sums, products, quotient.
Comparison test, Squeeze Theorem for sequences.
Monotone sequences. Important result: any bounded monotone sequence is convergent.
Cauchy sequence. Theorem: A sequence in R is convergent if and only if it is Cauchy. (Cauchy principle of convergence).
|Week 3||Proof of Cauchy principle of convergence.
The Bolzano-Weierstrass Theorem. Important Theorem. Equivalence of Completeness and Cauchy Principle of convergence and the Conclusion of Bolzano Weierstrass Theorem. Subsequences. Technical result: any sequence has a monotone subsequence. Proof of the Bolzano-Weierstrass Theorem.
|Week 4||Series. Definition of n-th partial sums
and convergence of a series. New meaning of the "=" sign to denote
the limit of the series. Example: Geometric series. Notation
and use of the summation sign.
Definition of a cauchy series. A series is convergent if and only if it is a cauchy series. (Cauchy principle of convergence). If ∑ an converges, then an → 0. Thus, if an does not converge to 0, then ∑ an diverges.
If ∑ an is a series of non-negative terms, then it is convergent if and only if it is bounded. Comparison test for series with non-negative terms. Example: ∑ 1/ n2 is convergent.
If ∑|an| is convergent, then ∑ an is convergent. Definition of absolute convergence. Examples.
∑ (-1)n+1 / n is convergent but not absolutely convergent. Definition of conditional convergence. Leibnitz"s Alternating series test. A very important test D' Alembert Ratio Test (for absolute convergence). A more refined version. Examples of absolute convergence and conditional convergence.
The Integral Test. The convergence and divergence for ∑ 1 / np for p > 0. Example of the use of comparison test . The Cauchy Root Test and a refined version using lim sup.
Example: ∑n=1n=∞ (-1)n+1/ n = ln(2).
|Week 5||Power Series. Definition and examples.
Domain of convergence, radius of convergence and the disk of convergence.
Proof of the basic theorem for series: it either converges only at 0, for
all x or converges absolutely inside a disk of radius r and diverges
outside a disk of radius r and may do either on the boundary circle of
convergence. Logarithmic series. The standard test for convergence is
still the ratio test. Examples of the use of this test for various power
Continuity of power series functions. It is continuous inside the (open) disk of convergence. For real power series, continuity at the end point of the interval of convergence is assured by Abel's Theorem it it is convergent there. Proof much later
|Week 6||Sequence of functions. Pointwise
convergence and uniform convergence of a sequence of functions.
Examples of pointwise convergence. Theorem If fn
converges uniformly to a function f on E, and if each fn
is continuous, then f is also continuous. Another proof of the continuity
of power series function using uniform convergence. Using the above
theorem to deduce non-uniform convergence. Examples.
Questions arising from the above theorem. Commutation of two different kinds of limiting process.
Determining radius of convergence. Using the standard Ratio Test . Using the Cauchy-Hadamard formula.
|Week 7||Weierstrass M-Test for uniform convergence of
a series of functions. Definition of a uniform Cauchy sequence of
functions. A sequence of (real valued) functions converges
uniformly if and only if it is uniformly Cauchy (Cauchy principle of
convergence) [ A sequence of functions in the space of bounded real valued
functions is convergent if and only if it is a Cauchy sequence with
respect to the sup metric. Ref: Binmore, Foundation of analysis Book 2
Topological Ideas.] Example of the use of the Weierstrass M-Test.
Example of a sequence of differentiable function fn converging
uniformly to a function f but the derivative fn ' does
not converge to f ' . We need
additional requirement on fn ' to deduce that
fn ' converges to f '. For this we can bring
the result about integrating a sequence of continuous function.
If gn is a sequence of continuous function converging uniformly to g on [a, b], then the sequence of integrals, ∫ gn from a to b converges to ∫ g from a to b. With this theorem we can then prove the following:
If fn converges pointwise to a function f and fn' converges uniformly to a function g, then f ' = g. Example: the exponential series. Application to power series function. The two power series obtained from a fixed one by differentiating or integrating its terms, term by term have the same radius of convergence as the given one. Consequence: we can differentiate a power series infinitely many times within its interval convergence.
Example of the use of Taylor's Theorem to find a solution to the differential equation f ' = f and f(0) =1. Example of a function that does not have a power series expansion. Condition of convergence of Taylor polynomials to give an infinite power series expansion fo a given function. A sufficient condition for Taylor series expansion. Example: the sine series expansion. Application: Proof that e is irrational.
Abels' Theorem and continuity of power series function at the boundary point of the disk of convergence.
|Week 8||Integration and uniform convergence.
Review of Riemann integration theory. Darboux sums, upper integral
and lower integrals. Characterization of Riemann integrability. Theorem:
the uniform limit of a sequence of Riemann integrable functions is
integrable. In particular the integral of the limiting function is
the limit of the sequence of integral of the term of the sequence.
Abel's test for the uniform convergence of infinite sums of series of
functions. Example. Dirichlet's test for the uniform
convergence of infinite sums of series of functions. Example: Simple
trigonometric series ∑ansin(nx), Power series expansion
for integrals with no closed form. Application: expansion for
A slight generalization of convergence theorem for sequence of integrals of functions, which form a sequence that converges only pointwise: The Arzela's Dominated Convergence Theorem. Application and examples.
Consequence of uniform convergence. Convergence of ∑ sin(nx)/ns, ∑ansin(nx), ∑an cos(nx). Newton's Binomial series.
|Week 9||Weierstrass Approximation Theorem.
Proof via Bernstein polynomial technique.
The elementary functions: exponential, logarithmic, sine and cosine functions defined analytically. Proof that they are the same as the usually defined functions. Properties and uniqueness of the sine and cosine functions. Definition of π (Richard Baltzer, Landau).
Sine and cosine thus defined and the trigonometric ratios.
Product of power series. Term wise product versus cauchy product.
Multiplication, quotient and inversion of Power series.
Special Tests for convergence of series. Kummer's Test, Raabe's Test, Gauss Test, Rabbe's type test and other more refined test.
Improper integrals. Definitions and examples. (For the first time Lebesgue integral is introduced) Improper Riemann integrals. Convergence Criteria and relation with Lebesgue integrals. Absolute and conditional convergence of improper Riemann integrals. Test for convergence. Examples.